How to get an arbitrary orientation of a graph.

I wish to take a simple undirected graph (i.e. the complete graph K_4)
Arbitrarily direct said graph, and then create a line graph from the directed version of the graph.

However, in Sage it appears to create a line graph that shows a connection between two edges (that are just inverses of each other), so what I really want is a line graph that doesn't give an edge connected to its own inverse.

That's why I asked if we could remove cycles of length 2, but that doesn't seem to solve the problem.

considering my graph is a complete graph on 4 vertices - the coefficients should be as such:
[coeff,degree of u]
[1,0], [0,1], [0,2],[-8,3],[-2,4]

NOTE:
im only interested in coefficients up to the order of n=#of nodes in the graph, so here for K_4 obviously n=4.
where the coefficient of u^3 corresponds to the negative of twice the number of triangles in K_4
where the coefficient u^4 corresponds to the negative of twice the number of squares in K_4

Here is an image of a K_4 graph minus an edge and the line graph construction of K_4 that i want

Comments

When I create a line graph from D as you defined above it doesn't give me representations of the directed edges inverses. I need the inverses of the arbitrarily oriented edges too, but obviously not having an inverse connected to its original edge. Anyway to do that?

maybe to clarify further: the to_directed() method was ok, as it gave me directed edges and their inverses, however i need a method in which the line graph recognises an edge's inverse and doesn't show a connection with the original edge. i.e. in the line graph edge(x,y) is not connected to edge (y,x) for all x,y in the edge set. Note: I added an image of what i'm after in the original post for further clarity